A celebrated result of Serre from 1972 asserts that if E/Q is an elliptic curve over Q without complex multiplication, then its associated mod l representation is surjective for any sufficiently large prime l. We will discuss how "sufficiently large" can be made effective in terms of the conductor of E. More precisely, we will explain the conditional (upon Riemann Hypothesis) approach given by Serre in 1981 ("Quelques applications du theoreme de densite de Chebotarev") and the uncoditional approach given by Kraus/Cojocaru.

No knowledge from the first lecture is assumed, as the techniques to be discussed are now analytic.